Trellis coded FM digital communications system and method

ABSTRACT

A method and apparatus for encoding and convolutionally decoding a data word of a data word sequence. For a data word having k bits, a 2 k  subset of signal points from a square set of M by M signal points having X coordinates and Y coordinates is selected using a trellis encoding algorithm and a previous data word. The square set of M by M signal points includes a constellation of 2 k+1  signal points. A signal point from the selected subset of signal points is selected using the n th  data word. A modulator generates a first FSK signal and a second FSK signal from the selected signal point. A demodulator demodulates the first FSK signal as a received X coordinate and the second FSK signal as a received Y coordinate. A 2 k  received subset of signal points is estimated by using the convolutionally decoding algorithm and a previously received data word. An n th  received data word is estimated from the received X coordinate, the received Y coordinate, and the estimated-received subset of signal points.

RELATED PATENTS

This patent is a continuation of a patent application entitled TRELLISCODED FM DIGITAL COMMUNICATIONS SYSTEM AND METHOD, having Ser. No.08/542,347 and filing date of Oct. 12, 1995 now U.S. Pat. No. 5,661,734,which is a continuation application of Ser. No. 08/261,062 and filingdate of Jun. 13, 1994, now U.S. Pat. No. 5,461,632 with issue date ofOct. 24, 1995, which is a continuation of Ser. No. 07/732,200 and filingdate of Jul. 19, 1991, now U.S. Pat. No. 5,351,249 with issue date ofSep. 27, 1994. The benefit of the earlier filing dates of the parentpatent applications is claimed pursuant to 35 U.S.C. § 120.

BACKGROUND OF THE INVENTION

This invention relates to a trellis encoded, narrowband FM system (NBFM)with spectral efficiency and low bit error rate.

DESCRIPTION OF THE RELEVANT ART

Trellis encoding has evolved primarily into phase and amplitudemodulation signalling schemes but not into frequency modulationsignalling schemes. A trellis code is a technique for encoding a datastream into a sequence of real vectors that are transmitted over a noisycommunications channel. Using phase modulation, two dimensional signalspaces are easily obtainable for generating vectors since in-phase andquadrature-phase signals can be generated with sine and cosine signals.Similarly, quadrature amplitude modulation (QAM) takes advantage ofin-phase and quadrature-phase signals, and varying amplitudes of asignal.

The mapping of two dimensional signal space is well developed in theprior art for PSK and QAM signalling schemes. A trellis code using PSKand QAM schemes increases power efficiency and reduces errors. Themapping of two dimensional signal spaces is not well developed, however,for FM systems.

OBJECTS OF THE INVENTION

A general object of the invention is to increase power efficiency ofM-ary digital FM communications using coding.

Another object of the invention is to relax the requirements on hardwarecapable of modulating and demodulating M-ary channel symbols while stillpermitting bandwidth efficiency, i.e., a high number of bits persymbols, communications. This objective is achieved by appealing to thefact that embedding messages in high dimensional signal spaces offersgreater immunity to noise and other system errors.

SUMMARY OF THE INVENTION

According to the present invention, as embodied and broadly describedherein, a trellis encoded, narrowband frequency modulation (NBFM)digital communications system is provided comprising first processormeans, modulator means, transmitter means, a communications channel,demodulator means, and second processor means. The system is used forencoding and decoding an n^(th) data word, or symbol, of a data wordsequence. Each encoded data word is assumed to have k bits per symbol.The system encodes the data word sequence, sends the encoded data wordsequence over the communications channel, and then decodes the data wordsequence.

More particularly, the first processor means is preprogrammed with asquare set of M by M signal points which has M indices of X coordinatesand M indices of Y coordinates. Alternatively, the first processor meansmay be preprogrammed with a rectangular set of signal points. A squareset of signal points is optimal. Additionally, the invention can beextended to Γ dimensions making a Γ dimensional cube.

The square set of M by M signal points is sufficiently large to includea constellation of 2^(k+1) signal points. The increase from 2^(k) to2^(k+1) represents the effect of coding. The first processor meansselects, using a trellis encoding algorithm and a previous data word ora plurality of previous data words, a 2^(k) subset of signal points fromthe square set of M by M signal points. The previous, (n-1)^(th), dataword is the data word that was processed previously in time to then^(th) data word presently being encoded. The processor means selects,using the n^(th) data word, a signal point from the selected subset ofsignal points. The selected signal point has an X coordinate and a Ycoordinate.

The modulator means can generate up to M channel symbols from an M-arysignalling scheme which correspond to the M indices of X coordinates andthe M indices of Y coordinates. The M-ary signalling scheme might have,for example, for six channel symbols, a six frequency-shift-keying (FSK)signalling scheme. Using the X coordinate and Y coordinate of theselected signal point, the modulator means generates a first channelsymbol of the M-ary signalling scheme from the X coordinate and a secondchannel symbol of the M-ary signalling scheme from the Y coordinate. Thetransmitter means transmits the first channel symbol and the secondchannel symbol over a communications channel. The first channel symboland the second channel symbol preferably are transmitted in sequence,although simultaneous multi-tone transmission is possible.

The demodulator means demodulates the first channel symbol as a receivedX coordinate and the second channel symbol as a received Y coordinate,thereby producing a two-dimensional demodulator output. Thetwo-dimensional demodulator outputs are input to the decoding algorithm,which is the Viterbi algorithm and which uses a previously received dataword, or a plurality of previously received data words, to produce amaximum likelihood sequence of estimates of the information symbols. Thesecond processor estimates an n^(th) received data word from thereceived X coordinate, the received Y coordinate, and the estimatedpreviously-received subset of signal points.

The present invention also may be a method for encoding and decoding ann^(th) data word of a data word sequence. The method comprises the stepsof selecting, using a predetermined coding algorithm and a previous dataword, a 2^(k) subset of signal points from a square set of M by M signalpoints having M indices of X coordinates and M indices of Y coordinates.The square set of M by M signal points includes a constellation of2^(k+1) signal points. The method also selects, using the n^(th) dataword, a signal point from the selected subset of signal points. A firstchannel symbol of an M-ary signalling scheme is generated from an Xcoordinate of the selected signal point, and a second channel symbol ofan M-ary signalling scheme is generated from a Y coordinate of theselected signal point. The first channel symbol and the second channelsymbol are transmitted over a communications channel.

The method includes demodulating the first channel symbol as a receivedX coordinate and the second channel symbol as a received Y coordinate,and estimating a 2^(k) received subset of signal points using thepredetermined algorithm and a previously received data word. An n^(th)received data word is estimated from the received X coordinate, thereceived Y coordinate, and the estimated-received subset of signalpoints.

Additional objects and advantages of the invention will be set forth inpart in the description which follows, and in part will be obvious fromthe description, or may be learned by practice of the invention. Theobjects and advantages of the invention also may be realized andattained by means of the instrumentalities and combinations particularlypointed out in the appended claims.

BRIEF DESCRIPTION OF THE DRAWINGS

The accompanying drawings, which are incorporated in and constitute apart of the specification, illustrate preferred embodiments of theinvention, and together with the description serve to explain theprinciples of the invention.

FIG. 1 is a block diagram of the trellis encoded, digital FMcommunications system;

FIG. 2A illustrates a code index scheme for a subset of signal points;

FIG. 2B illustrates a square set of signal points for M=6;

FIG. 3A is an example of a first selected signal point within a firstsubset of signal points;

FIG. 3B is an example of a second selected signal point within a secondsubset of signal points; and

FIG. 3C is an example of a third selected signal point within a thirdsubset of signal points.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

Reference will now be made in detail to the present preferredembodiments of the invention, examples of which are illustrated in theaccompanying drawings, wherein like reference numerals indicate likeelements throughout the several views.

In the exemplary arrangement shown in FIG. 1, a trellis encoded, digitalcommunications system is provided comprising first processor means,modulator means, transmitter means, a communications channel 50,demodulator means, and second processor means. The first processor meansis embodied as a first processor 22, the modulator means is embodied asa modulator 24 and the transmitter means is embodied as a transmitter26. The first processor 22 is coupled to a data source. The modulator 24is coupled between the first processor 22 and the transmitter 26, andthe transmitter 26 is coupled to the communications channel 50.

The demodulator means broadly may be embodied as a demodulator 32. Fornarrowband frequency modulation (NBFM) using FSK, the modulator 24 mayinclude an FM or FSK modulator, such as a voltage-controlled oscillatoror Armstrong type of modulator circuit, and the demodulator 32 mayinclude an FM or FSK demodulator, such as an FM discriminator, aphase-locked-loop (PLL) circuit or other FM demodulator circuit. Thesecond processor means may be embodied as a second processor 34. Thesecond processor 34 is coupled between the demodulator 32 and a dataoutput. The first processor 22 may be separate from a trellis encoder18, or the first processor 22 may include the trellis encoder 18.Similarly, the second processor 34 may be separate from a trellisdecoder 36, or the second processor 34 may include the trellis decoder36.

The digital communications system is used for encoding and decoding adata word of a data word sequence. The invention is taught, by way ofexample, with encoding and decoding of an n^(th) data word. Each dataword is assumed to have k bits. The system encodes the data wordsequence, sends the encoded data word sequence over the communicationschannel 50 using NBFM FSK modulation, and then decodes the data wordsequence.

More particularly, the first processor 22 is preprogrammed with a squareset of M by M signal points which has M indices of coordinates and Mindices of Y coordinates. Alternatively, the first processor means maybe preprogrammed with a rectangular set of signal points. A square setof signal points is optimal. Additionally, the invention can be extendedto Γ dimensions making a Γ dimensional cube.

The square set of M by M signal points includes a constellation of2^(k+1) signal points. For data words having four (k=4) data bits, byway of example, the square set would include six by six signal points,which is a square set of 36 signal points. The 36 signal points includethe constellation of 2⁴⁺¹, which equals 32, signal points.

The first processor 22, using a trellis encoding algorithm and aprevious data word or a plurality of previous data words, selects a2^(k) subset of signal points from the square set of M by M signalpoints. As an example, the subset of signal points may include four byfour, which equals a square set of sixteen, signal points. In practice,rectangular, circular or any other constellation of a subset of signalpoints can be used with the present invention. Trellis encodingalgorithms, which are well known in the art for PSK and QAM signallingschemes, can be used with the FM encoding scheme disclosed herein.

The previous data word and each of the plurality of previous data wordsis a data word that was processed previously in time to the n^(th) dataword presently being encoded. Thus, the (n-1)^(th) data word would beused by the predetermined algorithm for selecting which four by foursignal points, in the example, are the subset of signal points from thesix by six signal points. The preferred encoding scheme would be basedon the coset codes of Calderbank and Sloane, which are based onlattice-type constellations. The scheme is well known in the art anddescribed in "New Trellis Codes Based on Lattices and Cosets," A. R.Calderbank and N. J. A. Sloane, IEEE TRANSACTIONS ON INFORMATION THEORY,Vol. IT-33, No. 2, March 1987.

Trellis coding schemes are also discussed in: U.S. Pat. No. 4,939,555having issue date of Jul. 3, 1990, and entitled TRELLIS CODINGARRANGEMENT to Calderbank and Sloane; U.S. Pat. No. 4,901,331 havingissue date of Feb. 13, 1990, and entitled TRELLIS CODES WITH PASS BANDSPECTRAL NULLS to Calderbank and Mazo; U.S. Pat. No. 4,831,635 havingissue date of May 16, 1989, and entitled TRELLIS CODES WITH SPECTRALNULLS to Lee and Calderbank; U.S. Pat. No. 4,788,694 having issue dateof Nov. 29, 1988, and entitled TRELLIS CODING WITH SUBSTATES (sic) toCalderbank; and U.S. Pat. No. 4,581,601 having issue date of Apr. 8,1986, and entitled MULTI-DIMENSIONAL CODING FOR ERROR REDUCTION toCalderbank and Sloane; all of these patents are hereby incorporatedherein by reference.

As an example, the previous signal point, which corresponds to theprevious data word, can be used for locating the subset of signal pointsin the square set of signal points. An algorithm could require that acorner of the subset of signal points always be the previous signalpoint.

Similarly, the concept can be extended to using a plurality of previousdata words for locating the subset of signal points in the square ofsignal points. An algorithm could require that a corner of the subset ofsignal points always be at a location determined from a function of thelocation of three previous signal points, for example.

The first processor 22 selects, using the n^(th) data word, a signalpoint from the selected subset of signal points. In the example, thedata bits of the four bit data word are split into two half-data wordsof two bits each by partitioning the four data bits in half. The firsthalf-data word selects a column from the four by four subset of datapoints, and the second half-data word selects a row from the four byfour subset of data points. The intersection of the selected row andcolumn is at the selected data point. The columns and rows of the subsetof signal points can be indexed with the binary numbers of all thepossible half-data words. While the ordering of the columns and rowsmust be consistent, they may be ordered according to binary numbering,Gray codes, or any other scheme.

FIG. 2A illustrates an ordering of columns and rows for a square subsetof signal points. Although a square subset of signal points is shown,other shapes such as rectangular, circular, or more generally anytwo-dimensional lattice, will work with the present invention.

The M by M signal points define M indices of X coordinates and M indicesof Y coordinates. The selected signal point has an X coordinate and a Ycoordinate, defined in the M by M signal points. The modulator cangenerate up to M channel symbols from an M-ary signalling scheme whichcorrespond to the M indices of X coordinates and the M indices of Ycoordinates.

FIG. 2B illustrates a square set of signal points for M=6, with thecoordinates, i.e., rows and columns labelled with the six FSKfrequencies. Thus, each of the X coordinates and the Y coordinates takeson one of the frequencies. The M-ary signalling scheme might have, forexample, for six channel symbols (M=6), a six frequency-shift-keying(FSK) signalling scheme. The FSK signals might modulate a carrier signalat a carrier frequency. Thus, for six FSK, the frequencies might be f₁=+5 kHz, f₂ =+3 kHz, f₃ =+1 kHz, f₄ =-1 kHz, f₅ =-3 kHz, and f₆ =-5 kHz,with respect to the carrier frequency.

Using the X coordinate and Y coordinate of the selected signal point,the modulator 24 generates a first channel symbol of the M-arysignalling scheme from the X coordinate and a second channel symbol ofthe M-ary signalling scheme from the Y coordinate. The combination ofthe first and second channel symbols is a code symbol. The modulator 24can modulate the first channel symbol and the second channel symbolsequentially or simultaneously in time.

The transmitter 26 transmits the first channel symbol and the secondchannel symbol over the communications channel 50. The first channelsymbol and the second channel symbol can be sent sequentially in time orsimultaneously in time. The first channel symbol is orthogonal to thesecond channel symbol. For the example, if the first channel symbol andthe second channel symbol are transmitted simultaneously, then thefrequencies, in the FSK signalling example, would be orthogonal to eachother. If the first channel symbol and the second channel symbol aretransmitted sequentially, then they are orthogonal because they aretransmitted at different points in time. Thus, if the first channelsymbol and the second channel symbol were +3 kHz and -5 kHz of an FSKsignalling scheme, then the transmitter would send a code symbol of +3kHz and -5 kHz, either sequentially in time or simultaneously in time,on the carrier frequency.

The demodulator 32 demodulates the first channel symbol as a received Xcoordinate and the second channel symbol as a received Y coordinate. Thereceived X coordinate and the received Y coordinate are voltages whichare noisy estimates of the X-Y coordinate points.

The second processor 34 estimates a 2^(k) received subset of signalpoints using a decoding algorithm and a previously received data word,or a plurality of previously received data words. Using a plurality ofpreviously received data words enables the second processor 34 to employinformation which spans the time period of many channel symbols in orderto make a decision on the n^(th) channel symbol.

The second processor 34 estimates an n^(th) received data word from thereceived X coordinate, the received Y coordinate, and theestimated-received subset of signal points. The decoding algorithm canbe based on known techniques such as those used for codes developed forphase-shift-keying (PSK) or quadrature amplitude modulation (QAM)schemes. The decoding algorithm in a preferred embodiment would be theViterbi algorithm, which is well known in the art.

The present invention alternatively may be a method for encoding anddecoding an n^(th) data word of a data word sequence. The methodcomprises the steps of selecting, using a trellis encoding algorithm anda previous data word, a 2^(k) subset of signal points from a square setof M by M signal points having X coordinates and Y coordinates. Thesquare set of M by M signal points includes a constellation of 2^(k+1)signal points. The method also selects, using the n^(th) data word, asignal point from the selected subset of signal points. A first channelsymbol of an M-ary signalling scheme is generated from an X coordinateof the selected signal point, and a second channel symbol of an M-arysignalling scheme is generated from a Y coordinate of the selectedsignal point. The first channel symbol and the second channel symbol aretransmitted over a communications channel.

The method also includes demodulating the first channel symbol as areceived X coordinate and the second channel symbol as a received Ycoordinate, and estimating a 2^(k) received subset of signal pointsusing the decoding algorithm and a previously received data word. Ann^(th) received data word is estimated from the received X coordinate,the received Y coordinate, and the estimated-received subset of signalpoints.

Operation of the invention can be understood by way of example, shown inFIGS. 3A-3C, using the ordering of columns and rows of a square subsetof signal points and the coordinate labelled with the FSK frequencies ofthe square set of signal points of FIGS. 2A and 2B, respectively. Assumea data bit sequence enters the first processor 22. Assume that the firstprocessor performs trellis encoding and outputs the following data bits:

    010101011111

Assume that the trellis encoder operates on data words of four bits perdata word, and the trellis encoder locates the subset of signal points,as indicated in the dashed area, sequentially as shown in FIGS. 3A-3C.For the first data word, i.e., the first four data bits, 0 1 0 1, afirst selected signal point is located in a first subset of signalpoints which intersect with the row and column of the square set ofsignal points at frequencies f₄ and f₃. Thus, frequencies f₄ and f₃ aresent over the communications channel.

For the second data word, i.e., the second four data bits, 0101, asecond selected signal point is located in a second subset of signalpoints which intersect with the row and column of the square set of thesignal points at frequencies f₂ and f₂. Thus, the frequencies f₂ and f₂are sent over the communications channel.

For the third data word, i.e., the third set of four data bits, 1 1 1 1,a third selected signal point is located in a third subset of signalpoints which intersect with the row and column of the square set ofsignal points at frequencies f₄ and f₆. Thus, the frequencies f₆ and f₄are sent over the communications channel.

At the receiver, the pairs of received frequencies, f₄ f₃, f₂ f₂ and f₄f₆, form a vector of received data points. The Viterbi algorithm can beused for trellis decoding this vector of data points, as is done forPSK, QAM, etc.

It will be apparent to those skilled in the art that variousmodifications can be made to the trellis coded FM communications systemand method of the instant invention without departing from the scope orspirit of the invention, and it is intended that the present inventioncover modifications and variations of communications using trellis codedFM provided they come in the scope of the appended claims and theirequivalents.

We claim:
 1. A system used with a communications channel for encodingand convolutionally decoding a plurality of data words in a data wordsequence, each of said plurality of data words having k bits, with kbeing a number of bits in each of said plurality of data words, and witheach of said plurality of data words defining a signal point,comprising:first processor means for selecting, for a first data word inthe data word sequence, using a trellis encoding algorithm and aprevious signal point, a first 2^(k) subset of signal points from atwo-dimensional lattice of signal points having a plurality of Xcoordinates and a plurality of Y coordinates, the two-dimensionallattice of signal points including a constellation of 2^(k+1) signalpoints, a corner of the first 2^(k) subset of signal points beingdefined by the previous signal point; said first processor means forselecting, using the first data word, a first signal point from thefirst 2^(k) subset of signal points, the first signal point having an Xcoordinate and a Y coordinate; a modulator for generating a firstfrequency-shift-keying (FSK) signal from an M-ary FSK signalling schemeof the X coordinate of said first signal point, and for generating asecond FSK signal from the M-ary FSK signalling scheme of the Ycoordinate of said first signal point; a transmitter for transmittingthe first FSK signal at a first frequency and the second FSK signal at asecond frequency over a communications channel as a first frequencypair, the first frequency being orthogonal to the second frequency; ademodulator for demodulating the first FSK signal as a first received Xcoordinate and the second FSK signal as a first received Y coordinate;and second processor means for estimating a first 2^(k) received subsetof signal points using a convolutionally decoding algorithm and apreviously received data word; said second processor means forestimating a first received data word from the first received Xcoordinate, the first received Y coordinate, and the first 2^(k)received subset of signal points; said first processor means forselecting, for a second data word in the data word sequence, using saidtrellis encoding algorithm and a previous signal point, a second 2^(k)subset of signal points from the two-dimensional lattice of signalpoints; said first processor means for selecting, using the second dataword, a second signal point from the second 2^(k) subset of signalpoints, the second signal point having an X coordinate and a Ycoordinate; said modulator for generating a third FSK signal from theM-ary FSK signalling scheme of the X coordinate of said second signalpoint, and for generating a fourth FSK signal from the M-ary FSKsignalling scheme of the Y coordinate of said second signal point; saidtransmitter for transmitting the third FSK signal at a third frequencyand the fourth FSK signal at a fourth frequency over a communicationschannel as a second frequency pair, the third frequency being orthogonalto the fourth frequency; said demodulator for demodulating the third FSKsignal as a second received X coordinate and the fourth FSK signal as asecond received Y coordinate; said second processor means for estimatinga second 2^(k) received subset of signal points using theconvolutionally decoding algorithm and a previously received data word;and said second processor means for estimating a second received dataword from the second received X coordinate, the second received Ycoordinate, and the second 2^(k) received subset of signal points. 2.The system as set forth in claim 1, with the corner of the first 2^(k)subset of signal points defined by a function of the previous signalpoint and a location of two signal points previous to the previoussignal point.
 3. The system as set forth in claim 1, with saidtransmitter transmitting the first FSK signal and the second FSK signalsequentially.
 4. The system as set forth in claim 1, with saidtransmitter transmitting the first FSK signal and the second FSK signalsimultaneously.
 5. The system as set forth in claim 1, with said firstprocessor means selecting the first 2^(k) subset of signal points from arectangular set of signal points.
 6. The system as set forth in claim 1,with said first processor means selecting the first 2^(k) subset ofsignal points from a circular set of signal points.
 7. The system as setforth in claim 1, with said first processor means selecting a first2^(k) subset of signal points from a square set of M by M signal points,with M being a number of indices of the plurality of X coordinates andthe plurality of Y coordinates.
 8. The system as set forth in claim 1,with said first processor means selecting the first 2^(k) subset ofsignal points using the trellis encoding algorithm with a plurality ofprevious signal points.
 9. The system as set forth in claim 1, with saidfirst processor means selecting the second 2^(k) subset of signal pointsusing the trellis encoding algorithm with a plurality of previous signalpoints.
 10. The system as set forth in claim 1, with said secondprocessor means estimating the first 2^(k) received subset of signalpoints using the convolutionally decoding algorithm and a plurality ofpreviously received data words.
 11. The system as set forth in claim 1,with said second processor means estimating the second 2^(k) receivedsubset of signal points using the convolutionally decoding algorithm anda plurality of previously received data words.
 12. The system as setforth in claim 1, said second processing means including a processor anda trellis decoder.
 13. The system as set forth in claim 1, said firstprocessing means including a processor and a trellis encoder.
 14. Thesystem as set forth in claim 1, said first processor means forselecting, for an nth data word in the data word sequence, using saidtrellis encoding algorithm and a previous signal point, an nth 2^(k)subset of signal points from the two-dimensional lattice of signalpoints;said first processor means for selecting, using the nth dataword, an nth signal point from the nth 2^(k) subset of signal points,the nth signal point having an X coordinate and a Y coordinate; saidmodulator for generating a fifth FSK signal from the M-ary FSKsignalling scheme of the X coordinate of said nth signal point, and forgenerating a sixth FSK signal from the M-ary FSK signalling scheme ofthe Y coordinate of said nth signal point; said transmitter fortransmitting the fifth FSK signal at a fifth frequency and the sixth FSKsignal at a sixth frequency over a communications channel as a thirdfrequency pair, the fifth frequency being orthogonal to the sixthfrequency; said demodulator for demodulating the fifth FSK signal as ann^(th) received X coordinate and the sixth FSK signal as an n^(th)received Y coordinate; said second processor means for estimating an nth2^(k) received subset of signal points using the convolutionallydecoding algorithm and a previously received data word; and said secondprocessor means for estimating an nth received data word from the nthreceived X coordinate, the n^(th) received Y coordinate, and the nth2^(k) received subset of signal points.
 15. The system as set forth inclaim 14, said convolutionally decoding algorithm for trellis decoding avector of received signal points formed by the first frequency pair, thesecond frequency pair, and the third frequency pair.
 16. The system asset forth in claim 14, with said first processor means selecting the nth2^(k) subset of signal points using the trellis encoding algorithm witha plurality of previous signal points.
 17. The system as set forth inclaim 14, with said second processor means estimating the nth 2^(k)received subset of signal points using the convolutionally decodingalgorithm and a plurality of previously received data words.